2sin (example 3.3)

Average Error: 36.9 → 0.4

Time: 15.5s

Precision: binary64

Cost: 26176

Math

\[\sin \left(x + \varepsilon\right) - \sin x \]

↓

\[\sin \varepsilon \cdot \cos x - \sin x \cdot \left(1 - \cos \varepsilon\right) \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

↓

(FPCore (x eps)
 :precision binary64
 (- (* (sin eps) (cos x)) (* (sin x) (- 1.0 (cos eps)))))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

↓

double code(double x, double eps) {
	return (sin(eps) * cos(x)) - (sin(x) * (1.0 - cos(eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) - (sin(x) * (1.0d0 - cos(eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

↓

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) - (Math.sin(x) * (1.0 - Math.cos(eps)));
}

Python

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

↓

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) - (math.sin(x) * (1.0 - math.cos(eps)))

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

↓

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) - Float64(sin(x) * Float64(1.0 - cos(eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

↓

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) - (sin(x) * (1.0 - cos(eps)));
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	36.9
Target	14.9
Herbie	0.4

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

1. Initial program 36.9

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Applied egg-rr 45.1

   \[\leadsto \color{blue}{\left(-1 - \left(\sin x - \sin \left(x + \varepsilon\right)\right)\right) + 1} \]

3. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x - \sin x \cdot \left(1 - \cos \varepsilon\right)} \]

4. Final simplification 0.4

   \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \left(1 - \cos \varepsilon\right) \]

Alternatives

Alternative 1
Error	15.1
Cost	26180

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \cos x\\ \mathbf{if}\;\varepsilon \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;t_0 - \left(1 - \cos \varepsilon\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x + t_0\right) - \sin x\\ \end{array} \]

Alternative 2
Error	16.1
Cost	19776

\[\sin \varepsilon \cdot \cos x - \left(1 - \cos \varepsilon\right) \cdot x \]

Alternative 3
Error	14.7
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.1
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 5
Error	28.7
Cost	6464

\[\sin \varepsilon \]

Alternative 6
Error	44.9
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))